Has there ever been a proof simplified by category theory? I need to know this f ...

Has there ever been a proof simplified by category theory? I need to know this for my undergraduate studies.


My temptation is to argue that this is mostly not the kind of benefit you should expect from learning category theory. Here are the sorts of benefits you actually get:It becomes easier to remember and reason about many definitions, because they become special cases of general category-theoretic definitions. For example, many interesting constructions in mathematics are adjoint functors.Once you know that a definition is a special case of a general category-theoretic definition, you can use other specializations of the general definition to figure out or guess how your definition behaves. For example, if you know that a functor you care about is a left adjoint, then you already know that it preserves colimits, but you also know that you have no reason to expect that it preserves limits.It becomes possible to ask certain kinds of questions which would otherwise be hard to make precise. For example, what is the Frobenius endomorphism really, and does there exist an analogue of it in characteristic zero? It's impossible to ask this question precisely until you know enough category theory to cleanly describe what the Frobenius endomorphism actually is, which is that it's a natural endomorphism of the identity functor on commutative FpFp-algebras. Now you can ask a precise question which you couldn't ask without knowing what natural transformations are: is there an interesting natural endomorphism of the identity functor on, say, all commutative rings? The answer is no, and you can prove this using the Yoneda lemma.Here's an example that might appear at the undergraduate level. One of the first bits of "category theory in other math" I encountered was Frobenius reciprocity (Induced representation) in representation theory, which I think is hard to make sense of without understanding what adjoint functors are. Once you understand adjoints, Frobenius reciprocity is just the statement that restriction is right adjoint to induction, and I think before you understand this statement, you haven't really understood what induced representations are. All you know is that it's this useful construction which has a few equivalent and equally mysterious definitions, but once you've gotten a bit more comfortable with the yoga of adjoint functors, Frobenius reciprocity tells you immediately that the induced representation construction can be writtenIndGH(V)=C[G]⊗C[H]V;IndHG(V)=C[G]⊗C[H]V;this is a special case of a more general result about restriction of modules.The point I really want to make here is that it's not that category theory helped me understand the proof of Frobenius reciprocity; in some sense, there's almost no proof to understand. It's that category theory helped me understand where the definition of the induced representation comes from in the first place.Here's an example that appears a bit higher than the undergraduate level. A characteristic class is a way of assigning to, say, a vector bundle a cohomology class in a particularly nice way; they're useful as a way of distinguishing vector bundles. You can write down a lot of characteristic classes more or less explicitly, but a nagging question remains: have you written down all of them, or are there more left to find?You can't answer this question until you get more specific about what a characteristic class actually is, and once you know enough category theory, you can say the right thing, which is that a characteristic class is a natural transformation between a vector bundle functor and a cohomology functor. Once you know, in addition, that the first functor is representable, then the Yoneda lemma tells you that a characteristic class is precisely a cohomology class on a classifying space. You've reduced a question that sounded like it was about all spaces to a question about a particular space, and in particular, once you compute the cohomology of this particular space you've found all characteristic classes, and you can rest assured that you haven't missed any. I think this argument is originally due to Serre (but the point is that I didn't need to be Serre to figure this out! I just needed to know what natural transformations and the Yoneda lemma are.)

Answered by Jaroslav

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